Derivation of transient relativistic fluid dynamics from the Boltzmann equation

Abstract

In this work we present a general derivation of relativistic fluid dynamics from the Boltzmann equation using the method of moments. The main difference between our approach and the traditional 14-moment approximation is that we will not close the fluid-dynamical equations of motion by truncating the expansion of the distribution function. Instead, we keep all terms in the moment expansion. The reduction of the degrees of freedom is done by identifying the microscopic time scales of the Boltzmann equation and considering only the slowest ones. In addition, the equations of motion for the dissipative quantities are truncated according to a systematic power-counting scheme in Knudsen and inverse Reynolds number. We conclude that the equations of motion can be closed in terms of only 14 dynamical variables, as long as we only keep terms of second order in Knudsen and/or inverse Reynolds number. We show that, even though the equations of motion are closed in terms of these 14 fields, the transport coefficients carry information about all the moments of the distribution function. In this way, we can show that the particle-diffusion and shear-viscosity coefficients agree with the values given by the Chapman-Enskog expansion.

Determining transport coefficients for a microscopic simulation of a hadron gas

“…We then fit the behavior of the stress-energy tensor from a microscopic simulation performed in a controlled environment by varying both η and γ π . Since this also fixes the relaxation time, τ π , we can compare the relaxation time to the collision time and see whether τ π ≈ 2τ coll , which was found for the case studied in [33,36].…”

“…(2) has an unknown dimensionless coefficient γ π . For kinetic theory of a massless gas with fixed cross sections and isotropic scattering, this coefficient is known to be 5/7 [33,36,37]. However, for this case we treat it as an unknown, so we are left with two free parameters to describe the evolution of π zz .…”

“…For a massless gas with fixed cross sections and isotropic scatterings, kinetic theory gives the coefficients as κ π = 4/3 and γ π = 5/7 [33,[36][37][38][39]. The coefficient κ π can be related to fluctuations of the stressenergy tensor, F π (see appendix).…”

“…For higher velocity gradients it becomes important to consider higher-order transport coefficient such as those found in Israel-Stewart theories. The application and extension of Israel-Stewart theories in [31][32][33][34] is especially relevant to heavy-ion collisions, where super-luminar modes can emerge if the equations are parabolic. Some terms for higher-order gradient expansions can also be addressed through Kubo relations [35].…”

Variational principle for theories with dissipation from analytic continuation

“…As discussed in appendix A, the Matsubara correlation function ∆ M ab is actually a special case of a more general correlation function G ab which is defined for general complex frequency argument. More specific, on the axis of imaginary Matsubara frequencies one has 17) and G ab (p 0 , p ) is the unique analytic continuation of ∆ M ab (iω n , p ) to the full plane of complex frequencies p 0 ∈ C which is analytic everywhere except for possible poles and brach cuts on the real frequency axis p 0 ∈ R. The complex argument Greens function has a Källen-Lehmann spectral representation which makes its analytic structure with respect to p 0 directly apparent, [47]. In momentum space,…”